On some subsets of the dual of a Banach space.

Abstract

With the definition that a Banach space E has the property sDP if the Dunford-Pettis operators and the unconditionally converging operators from E into F coincide for every Banach space F, the author proves that E has property sDP if and only if two specified classes of subsets of the dual space E\sp* of E coincide. He obtains a corresponding characterization of the Dunford-Pettis property of a Banach space E, i.e., that every weakly compact operator from E into F is also a Dunford-Pettis operator for every Banach space F. Additional results about the property sDP and easy proofs of certain known theorems are also given.